10.6 Variational Problems of Functionals with Parallel-Type Inner …
705
The trace of a tensor can be expressed as tr A = I n
· A, its variation is δtr A =
I n
· δ A, here I is the unit tensor of order n.
Making use of the concepts of the above-mentioned Hamiltonian operator trains,
the more complicated variational problem can be discussed.
Theorem 10.6.2 Let A be a tensor of order n, ∇ t is a left Hamiltonian operator
train, ∇
a
t is the left conjugate Hamiltonian operator train, t ∇ is a right Hamiltonian
operator train,
a
t ∇ is the right conjugate Hamiltonian operator train, if the functional
with the tensor and the Hamiltonian operator trains can be expressed as
J =
V
F( A, A
T
, tr A, ∇ t A t ∇)dV
(10.6.43)
then the Euler equation of the functional is
∂ F
∂ A
+
∂ F
∂ A
T
+ I + (−1)
k
∇
c
t
∂ F
∂∇ t A t ∇
c
t ∇ = 0
(10.6.44)
where, k is the sum of the gradient operator quantities and the divergence operator
quantities in the left and right Hamiltonian operator trains, in this case, the sign of
the adjoint Hamiltonian operator train is not considered any more. The method of
adding brackets can be used to determine the operational sequence of the Hamiltonian
operator trains. If the brackets are not added, then first the left Hamiltonian operator
train is calculated, then the right Hamiltonian operator train is calculated, or the
operational sequence is determined according to reality.
Proof Only the last term of Eq. (10.6.44) needs to be proved.
When taking the variation to ∇ t A, for the Hamiltonian operator train ∇ t , every
time only one Hamiltonian operator can be reserved on the left-hand side of the
variational symbol δ, the right-hand side of δ is regarded as a tensor, according to
the leftmost Hamiltonian operator being the gradient, the divergence or the rotation,
respectively substitute ∇ t δ A into the above corresponding variational formulas, to
make the Hamiltonian operators in ∇ t δ A reduce, the left-hand side of ∇ t δ A every
time reduces a gradient operator, the left-hand side of its adjoint tensor increases a
divergence operator, the left-hand side of ∇ t δ A every time reduces a divergence operator, the left-hand side of its adjoint tensor increases a gradient operator, the left-hand
side of ∇ t δ A every time reduces a rotation operator, the left-hand side of its adjoint
tensor increases a rotation operator, finally δ∇ t A can be changed into the form of δ A,
meanwhile the corresponding boundary conditions can be obtained too. Similarly,
δ A t ∇ can also be changed into the form of δ A. Notice that the gradient operator and
the divergence operator are operated one time, their signs are just changed one time,
but the rotation operator does not change its sign in the operation process. Therefore
the last term of Eq. (10.6.44) can be obtained. Quod erat demonstrandum.
705
The trace of a tensor can be expressed as tr A = I n
· A, its variation is δtr A =
I n
· δ A, here I is the unit tensor of order n.
Making use of the concepts of the above-mentioned Hamiltonian operator trains,
the more complicated variational problem can be discussed.
Theorem 10.6.2 Let A be a tensor of order n, ∇ t is a left Hamiltonian operator
train, ∇
a
t is the left conjugate Hamiltonian operator train, t ∇ is a right Hamiltonian
operator train,
a
t ∇ is the right conjugate Hamiltonian operator train, if the functional
with the tensor and the Hamiltonian operator trains can be expressed as
J =
V
F( A, A
T
, tr A, ∇ t A t ∇)dV
(10.6.43)
then the Euler equation of the functional is
∂ F
∂ A
+
∂ F
∂ A
T
+ I + (−1)
k
∇
c
t
∂ F
∂∇ t A t ∇
c
t ∇ = 0
(10.6.44)
where, k is the sum of the gradient operator quantities and the divergence operator
quantities in the left and right Hamiltonian operator trains, in this case, the sign of
the adjoint Hamiltonian operator train is not considered any more. The method of
adding brackets can be used to determine the operational sequence of the Hamiltonian
operator trains. If the brackets are not added, then first the left Hamiltonian operator
train is calculated, then the right Hamiltonian operator train is calculated, or the
operational sequence is determined according to reality.
Proof Only the last term of Eq. (10.6.44) needs to be proved.
When taking the variation to ∇ t A, for the Hamiltonian operator train ∇ t , every
time only one Hamiltonian operator can be reserved on the left-hand side of the
variational symbol δ, the right-hand side of δ is regarded as a tensor, according to
the leftmost Hamiltonian operator being the gradient, the divergence or the rotation,
respectively substitute ∇ t δ A into the above corresponding variational formulas, to
make the Hamiltonian operators in ∇ t δ A reduce, the left-hand side of ∇ t δ A every
time reduces a gradient operator, the left-hand side of its adjoint tensor increases a
divergence operator, the left-hand side of ∇ t δ A every time reduces a divergence operator, the left-hand side of its adjoint tensor increases a gradient operator, the left-hand
side of ∇ t δ A every time reduces a rotation operator, the left-hand side of its adjoint
tensor increases a rotation operator, finally δ∇ t A can be changed into the form of δ A,
meanwhile the corresponding boundary conditions can be obtained too. Similarly,
δ A t ∇ can also be changed into the form of δ A. Notice that the gradient operator and
the divergence operator are operated one time, their signs are just changed one time,
but the rotation operator does not change its sign in the operation process. Therefore
the last term of Eq. (10.6.44) can be obtained. Quod erat demonstrandum.